How can I heal an ingrown toenail swollen area?

It may hurt, but... It's possible but unlikely that it will just heal. There is a small piece of nail still embedded in the corner. You need to get that out. If you can't get it out on your own, a podiatrist can. Sometimes this can be done without numbing the toe, but often, anesthesia is necessary. If it is a chronic condition, there is a very successful surgery that can be done to permanently correct the problem.
Nail care. The ingrown part of the nail is not always visible. Try soaking your foot in warm water, removal of embedded debris from the nail borders, and applying an antibiotic ointment to temporarily soften the corners. If you have redness and drainage, get it looked at by a professional, as you may need to be prescribed antibiotics and you need to have the ingrown nail border removed.
See foot doc. Sometimes antibiotic may be sufficient, other times need partial nail avulsion.